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Lesson 10.1 • The Geometry of Solids
Name
Period
Date
For Exercises 1–14, refer to the figures below.
T
O
A
I
D
H
C
J
E
P
R
Q
B
B
C
G
F
A
1. The cylinder is (oblique, right).
is ________________ of the cylinder.
2. OP
is ________________ of the cylinder.
3. TR
4. Circles O and P are ________________ of the cylinder.
is ________________ of the cylinder.
5. PQ
6. The cone is (oblique, right).
7. Name the base of the cone.
8. Name the vertex of the cone.
9. Name the altitude of the cone.
10. Name a radius of the cone.
11. Name the type of prism.
12. Name the bases of the prism.
13. Name all lateral edges of the prism.
14. Name an altitude of the prism.
In Exercises 15–17, tell whether each statement is true or false. If the
statement is false, give a counterexample or explain why it is false.
15. The axis of a cylinder is perpendicular to the base.
16. A rectangular prism has four faces.
17. The bases of a trapezoidal prism are trapezoids.
For Exercises 18 and 19, draw and label each solid. Use dashed lines to
show the hidden edges.
18. A right triangular prism with height
19. An oblique trapezoidal pyramid
equal to the hypotenuse
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Lesson 10.2 • Volume of Prisms and Cylinders
Name
Period
Date
In Exercises 1–3, find the volume of each prism or cylinder.
All measurements are in centimeters. Round your answers to the
nearest 0.01.
1. Right triangular prism
2. Right trapezoidal prism
6
6
6
14
10
3. Regular hexagonal prism
8
4
10
5
3
In Exercises 4–6, use algebra to express the volume of each solid.
4. Right rectangular prism
5. Right cylinder;
base circumference p
6. Right rectangular prism
and half of a cylinder
4y
y
x
h
2x 3
2x
3x
7. You need to build a set of solid cement steps for the entrance
6 in.
to your new house. How many cubic feet of cement do
you need?
3 ft
8 in.
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Lesson 10.3 • Volume of Pyramids and Cones
Name
Period
Date
In Exercises 1–3, find the volume of each solid. All measurements are in
centimeters. Round your answers to two decimal places.
1. Rectangular pyramid; OP 6 2. Right hexagonal pyramid
3. Half of a right cone
P
25
9
O
14
5
8
6
In Exercises 4–6, use algebra to express the volume of each solid.
4.
6. The solid generated by
5.
30x
b
spinning ABC about
the axis
2a
A
7x
25x
3x
C
2y
B
In Exercises 7–9, find the volume of each figure and tell which volume is larger.
7.
A.
B.
4
12
8
6
8.
A.
B.
2
3
3
5
5
2
9.
A.
B.
x
x
3
9
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Lesson 10.4 • Volume Problems
Name
Period
Date
1. A cone has volume 320 cm3 and height 16 cm. Find the radius of the
base. Round your answer to the nearest 0.1 cm.
2. How many cubic inches are there in one cubic foot? Use your answer
to help you with Exercises 3 and 4.
3. Jerry is packing cylindrical cans with diameter 6 in. and height 10 in.
tightly into a box that measures 3 ft by 2 ft by 1 ft. All rows must
contain the same number of cans. The cans can touch each other. He
then fills all the empty space in the box with packing foam. How many
cans can Jerry pack in one box? Find the volume of packing foam he
uses. What percentage of the box’s volume is filled by the foam?
4. A king-size waterbed mattress measures 72 in. by 84 in. by 9 in. Water
weighs 62.4 pounds per cubic foot. An empty mattress weighs
35 pounds. How much does a full mattress weigh?
5. Square pyramid ABCDE, shown at right, is cut out of a cube
E
. AB 2 cm. Find the volume
with base ABCD and shared edge DE
and surface area of the pyramid.
6. In Dingwall the town engineers have contracted for a new water
storage tank. The tank is cylindrical with a base 25 ft in diameter
and a height of 30 ft. One cubic foot holds about 7.5 gallons of
water. About how many gallons will the new storage tank hold?
C
D
A
B
7. The North County Sand and Gravel Company stockpiles sand
to use on the icy roads in the northern rural counties of the
state. Sand is brought in by tandem trailers that carry 12 m3
each. The engineers know that when the pile of sand, which is in
the shape of a cone, is 17 m across and 9 m high they will have
enough for a normal winter. How many truckloads are needed to
build the pile?
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Lesson 10.5 • Displacement and Density
Name
Period
Date
1. A stone is placed in a 5 cm-diameter graduated cylinder, causing the
water level in the cylinder to rise 2.7 cm. What is the volume of
the stone?
2. A 141 g steel marble is submerged in a rectangular prism with base
5 cm by 6 cm. The water rises 0.6 cm. What is the density of the steel?
3. A solid wood toy boat with a mass of 325 g raises the water level of a
50 cm-by-40 cm aquarium 0.3 cm. What is the density of the wood?
4. For Awards Night at Baddeck High School, the math club is
designing small solid silver pyramids. The base of the pyramids will
be a 2 in.-by-2 in. square. The pyramids should not weigh more than
212 pounds. One cubic foot of silver weighs 655 pounds. What is the
maximum height of the pyramids?
5. While he hikes in the Gold Country of northern California, Sid
dreams about the adventurers that walked the same trails years ago.
He suddenly kicks a small bright yellowish nugget. Could it be gold?
Sid quickly makes a balance scale using his walking stick and finds that
the nugget has the same mass as the uneaten half of his 330 g nutrition
bar. He then drops the stone into his water bottle, which has a 2.5 cm
radius, and notes that the water level goes up 0.9 cm. Has Sid struck
gold? Explain your reasoning. (Refer to the density chart in
Lesson 10.5 in your book.)
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Lesson 10.6 • Volume of a Sphere
Name
Period
Date
In Exercises 1–6, find the volume of each solid. All measurements are in
centimeters. Write your answers in exact form and rounded to the nearest
0.1 cm3.
1.
2.
3.
6
3
6
4.
5.
2
6
6. Cylinder with hemisphere
taken out of the top
6
90°
6
9
4
5
7. A sphere has volume 2216 cm3. What is its diameter?
8. The area of the base of a hemisphere is 225 in2. What is its volume?
9. Eight wooden spheres with radii 3 in. are packed snugly into a square
box 12 in. on one side. The remaining space is filled with packing
beads. What is the volume occupied by the packing beads? What
percentage of the volume of the box is filled with beads?
10. The radius of Earth is about 6378 km, and the radius of Mercury is
about 2440 km. About how many times greater is the volume of Earth
than that of Mercury?
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Lesson 10.7 • Surface Area of a Sphere
Name
Period
Date
In Exercises 1–4, find the volume and total surface area of each solid.
All measurements are in centimeters. Round your answers to the nearest
0.1 cm.
1.
2.
4
7
7.2
3.
4.
5
8
3
3
3
5. If the surface area of a sphere is 48.3 cm2, find its diameter.
6. If the volume of a sphere is 635 cm3, find its surface area.
7. Lobster fishers in Maine often use spherical buoys to mark their lobster
traps. Every year the buoys must be repainted. An average buoy has a
12 in. diameter, and an average fisher has about 500 buoys. A quart of
marine paint covers 175 ft2. How many quarts of paint does an average
fisher need each year?
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LESSON 9.6 • Circles and the Pythagorean Theorem
1. (25 24) cm2, or about 54.5 cm2
2. (723 24) cm2, or about 49.3 cm2
3. (5338
37) cm 36.1 cm
4. Area 56.57 cm 177.7 cm2
7. 150°
LESSON 10.1 • The Geometry of Solids
2. 209.14 cm3 3. 615.75 cm3
8
4. V 840x3
5. V 3a 2b 6. V 4xy 2
7. A: 128 cubic units, B: 144 cubic units.
B is larger.
9. A: 9x cubic units, B: 27x cubic units. B is larger.
LESSON 10.4 • Volume Problems
1. oblique
2. the axis
3. the altitude
4. bases
5. a radius
6. right
7. Circle C
8. A
or AC
9. AC
or BC
10. BC
1. 80 cm3
8. A: 5 cubic units, B: 5 cubic units.
They have equal volumes.
5. AD 115.04
cm 10.7 cm
6. ST 93 15.6
LESSON 10.3 • Volume of Pyramids and Cones
11. Right pentagonal prism
1. 4.4 cm
2. 1728 in3
3. 24 cans; 3582 in3 2.07 ft3; 34.6%
4. 2000.6 lb (about 1 ton)
12. ABCDE and FGHIJ
AB
and EC
BC
. V 8 cm3;
5. Note that AE
3
SA (8 42) cm2 13.7 cm2
, BG
, CH
, DI
, EJ
13. AF
6. About 110,447 gallons
, BG
, CH
, DI
, EJ
or their lengths
14. Any of AF
7. 57 truckloads
15. False. The axis is not perpendicular to the base in
an oblique cylinder.
16. False. A rectangular prism has six faces. Four are
called lateral faces and two are called bases.
17. True
18.
LESSON 10.5 • Displacement and Density
All answers are approximate.
1. 53.0 cm3
2. 7.83 g/cm3
3. 0.54 g/cm3
4. 4.94 in.
5. No, it’s not gold (or at least not pure gold). The
mass of the nugget is 165 g, and the volume is
17.67 cm3, so the density is 9.34 g/cm3. Pure gold
has density 19.3 g/cm3.
LESSON 10.6 • Volume of a Sphere
19.
1. 288 cm3, or about 904.8 cm3
2. 18 cm3, or about 56.5 cm3
LESSON 10.2 • Volume of Prisms and Cylinders
1. 232.16 cm3
2. 144 cm3
3. 415.69 cm3
4. V 4xy(2x 3), or 8x 2y 12xy
1
1
5. V 4p 2h
6. V 6 2x 2y
7. 6 ft3
3. 72 cm3, or about 226.2 cm3
28
4. 3 cm3, or about 29.3 cm3
5. 432 cm3, or about 1357.2 cm3
304
6. 3 cm3, or about 318.3 cm3
7. 11 cm
8. 2250 in 3 7068.6 in 3
9. 823.2 in3; 47.6%
10. 17.86
LESSON 10.7 • Surface Area of a Sphere
1. V 1563.5 cm3; S 651.4 cm2
2. V 184.3 cm3; S 163.4 cm2
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ANSWERS
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3. V 890.1 cm3; S 486.9 cm2
6. CA 64 cm
4. V 34.1
7. ABC EDC. Possible explanation: A E
and B D by AIA, so by the AA Similarity
Conjecture, the triangles are similar.
cm3;
S 61.1
cm2
5. About 3.9 cm
6. About 357.3 cm2
8. PQR STR. Possible explanation: P S
and Q T because each pair is inscribed in the
same arc, so by the AA Similarity Conjecture, the
triangles are similar.
7. 9 quarts
LESSON 11.1 • Similar Polygons
1. AP 8 cm; EI 7 cm; SN 15 cm; YR 12 cm
2. SL 5.2 cm; MI 10 cm; mD 120°;
mU 85°; mA 80°
3. Yes. All corresponding angles are congruent. Both
figures are parallelograms, so opposite sides within
each parallelogram are equal. The corresponding
sides are proportional 155 93.
4. Yes. Corresponding angles are congruent by the CA
Conjecture. Corresponding sides are proportional
2
3
4
4 = 6 = 8.
6
8
5. No. 1
8 22 .
6. Yes. All angles are right angles, so corresponding
angles are congruent. The corresponding side
lengths have the ratio 47, so corresponding side
lengths are proportional.
1
7. 2
y
4
A(0, 1)
LESSON 11.3 • Indirect Measurement with
Similar Triangles
1. 27 ft
2. 6510 ft
4. About 18.5 ft
5. 0.6 m, 1.2 m, 1.8 m, 2.4 m, and 3.0 m
LESSON 11.4 • Corresponding Parts of Similar Triangles
1. h 0.9 cm; j 4.0 cm
2. 3.75 cm, 4.50 cm, 5.60 cm
5
3. WX 137 13.7 cm; AD 21 cm; DB 12 cm;
6
YZ 8 cm; XZ 67 6.9 cm
50
80
4. x 1
3 3.85 cm; y 13 6.15 cm
6. CB 24 cm; CD 5.25 cm; AD 8.75 cm
C(1.5, 1.5)
D(2, 0.5)
x
4
LESSON 11.5 • Proportions with Area
1. 5.4 cm2
2. 4 cm
25
5. 4
6. 16:25
9. 1296 tiles
y
D (2, 4)
5
x
LESSON 11.2 • Similar Triangles
1. MC 10.5 cm
2. Q X; QR 4.8 cm; QS 11.2 cm
3. A E; CD 13.5 cm; AB 10 cm
4. TS 15 cm; QP 51 cm
5. AA Similarity Conjecture
ANSWERS
9
3. 2
5
7. 2:3
36
4. 1
8
8. 8889 cm2
LESSON 11.6 • Proportions with Volume
E(8, 2)
F(4, 2)
112
3. 110.2 mi
5. a 8 cm; b 3.2 cm; c 2.8 cm
B (2, 3)
8. 4 to 1
5
9. MLK NOK. Possible explanation:
MLK NOK by CA and K K because
they are the same angle, so by the AA Similarity
Conjecture, the two triangles are similar.
3. 16 cm3
1. Yes
2. No
5. 8:125
6. 6 ft2
4. 20 cm
LESSON 11.7 • Proportional Segments Between
Parallel Lines
1. x 12 cm
2. Yes
3. No
4. NE 31.25 cm
5. PR 6 cm; PQ 4 cm; RI 12 cm
6. a 9 cm; b 18 cm
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